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We discuss the equivalence relation between the Euclidean bipartite matchingproblem on the line and on the circumference and the Brownian bridge process onthe same domains. The equivalence allows us to compute the correlation functionand the optimal cost of the original combinatoric problem in the thermodynamiclimit; moreover, we solve also the minimax problem on the line and on thecircumference. The properties of the average cost and correlation functions arediscussed.

On the one dimensional Euclidean matching problem: exact solutions, correlation functions and universality